Operator valued analytic functions on an annulus

Question

Let $\mathscr{L, M}$ be two Hilbert spaces (not necessarily finite dimensional) and let $$\mathbb{A}=\{z\in \mathbb{C}:0<q<|z|<1\}$$ be an annulus. I was trying to learn about $B(\mathscr{L, M})$-valued (bounded operators from $\mathscr{L}$ to $\mathscr{M}$) analytic functions on $\mathbb{A}$. But all I found was literature on $B(\mathbb{C}^n, \mathbb{C}^m)$-valued functions with finite $m$ and $n$. Can anyone give me some reference on this topic? Specially on the infinite dimensional case.

Answer 1

For any open set $\Omega \subset \mathbb{C}$ and complex Frechet space $X$ a function $f \colon \Omega \rightarrow X$ is holomorphic if and only if it is weakly holomorphic, see Theorem 3.31, Chapter 1, in "Functional Analysis" by Rudin. Weakly holormophic means here that for any $\Lambda \in X^*$ the function $\Lambda(f)$ is holomorphic. Moreover, we also know that the Cauchy formula holds. Thus $f$ has Laurent series representation. (Similarly, we can show other properties of $f$ by using this statements for $\Lambda(f)$. For example, if $f(z)=0$ then also $\Lambda(f)(z) =0$. Assuming that $f$ is non-trival, we see that there exists (by Hahn-Banach) $\Lambda \in X^*$ with $\Lambda(f) \neq 0$. Thus, the zeros of $f$ don't have any accumulation point in $\mathbb{A}$ and this set is countable.)

However, I don't know any reference on this special subject. Which questions do you have exactly? Additionally, the statements on $B(\mathbb{C}^n,\mathbb{C}^m)$ may be extended to the infinite dimensional case.